Questions: f(x) = log(sqrt(2))(x^2 - 4)

Solution Steps

To solve the function \( f(x) = \log_{\sqrt{2}}(x^2 - 4) \), we need to evaluate the logarithm with base \(\sqrt{2}\) for the expression \(x^2 - 4\). We can use the change of base formula for logarithms to convert the base to a more convenient one, such as base 10 or base \(e\).

1. Use the change of base formula: \(\log_b(a) = \frac{\log_k(a)}{\log_k(b)}\), where \(k\) is a new base (commonly 10 or \(e\)).
2. Substitute \(b = \sqrt{2}\) and \(a = x^2 - 4\).
3. Implement the formula in Python to evaluate the function for given values of \(x\).

We start with the function defined as: \[ f(x) = \log_{\sqrt{2}}(x^2 - 4) \] To evaluate this function, we need to ensure that the argument \(x^2 - 4\) is positive, which requires \(x^2 > 4\) or \(|x| > 2\).

Substituting \(x = 5\) into the function: \[ f(5) = \log_{\sqrt{2}}(5^2 - 4) = \log_{\sqrt{2}}(25 - 4) = \log_{\sqrt{2}}(21) \]

Using the change of base formula: \[ \log_{\sqrt{2}}(21) = \frac{\log(21)}{\log(\sqrt{2})} \] Calculating the logarithms, we find: \[ \log(21) \approx 1.3222 \quad \text{and} \quad \log(\sqrt{2}) = \frac{1}{2} \log(2) \approx 0.3466 \] Thus, \[ f(5) \approx \frac{1.3222}{0.3466} \approx 3.81 \]

Final Answer